The Rise and Fall of Pibals

by Phil Lowell and Joe Reynolds

What, another article on the speed at which pibals
rise? Wasn't there recently an article by Mike Bien [Balloon Life,
page 18, July 95] that told more than we wanted to know about them? Well,
yes, but here is the new twist. We did some experimental work and by making
a force balance we derived a magic formula that only requires you to get
an average weight for your balloons and to measure the "diameter"
before you release them. You can then look up the balloon rise speed using
the figure or table in this article. An important point of this article
is that the balloon rise speeds given here are based on pibals that balloon
pilots use.

Just to give you a little background, engineers
have been calculating the drop rate of spheres for decades - or is it centuries?
All of this experimental work is boiled down to one number, a drag coefficient.
Now a balloon is almost like a sphere, right? And gravity pushing a steel
or polyethylene sphere down through air or water should be no different
than gravity pushing a sphere up through air, right? Wrong!

Recently some learned folks have measured (instead
of calculating the obviously already known solution) the rise rate of light
spheres - like a pibal in air. They found the drag coefficients were greater
by a factor of two or more [see D. G. Karamanev, et al, AIChEJ vol 42, 1789-92
(1996)]. The reason put forth for this discrepancy is that in the early
experiments, spheres of steel, nylon, glass, etc. were used. These were
heavy spheres and had a lot in inertia to resist twisting (a large angular
moment of inertia for you science buffs). The pibals, on the other hand
have a low moment of angular inertia. The drag forces exerted on the pibal
cause it to twist and turn. Actually it will often corkscrew (spiral) up.
This changes the effective drag coefficient. So much for talking about background,
here is what we did.

We ran two sets of experiments. We first weighed
the balloons empty. For the first set we filled them with welder's helium
and tied them off. We clipped a binder clip on the tie-off and weighed the
balloon. We put the balloon between two carpenter's squares and measured
the diameter and "height" of the balloon. We call the balloon
"tie-off" the bottom. The height is the distance between the bottom
and the other end - which we call the top. We call the diameter the largest
circle of balloon perpendicular to the bottom-top axis.

Now for the action. Inside a four story office
building with a 69 foot atrium ceiling we released the balloons and timed
them to hitting the glass ceiling. This was done after working hours when
the AC was off.

The second set of experiments we ran with a mixture
of helium and air - just for science. As an aside, welder's helium is very
close to being pure helium. Party balloon helium is often an air/helium
mixture. The air keeps the balloons from deflating and looking sick as the
helium diffuses out. The party balloon helium filled balloons look OK, they
just may not rise at a predictable speed. At any rate, we timed how fast
these balloons hit the ceiling. For the sake of pure science, we filled
some balloons with air and dropped them from the fourth floor and timed
their descent (OK, so we ran out of helium before we ran out of balloons
and were having too much fun wondering what folks would say Monday morning
when they found the atrium ceiling covered with balloons).

For a sphere the lift would be the volume of the
sphere times the density difference between air and helium, Pair
- PHeminus the weight of the balloon itself. Since a balloon
is not a perfect sphere, we throw in a shape factor, k, for the volume of
the sphere.

Vballoon = k(/6)D3

Now for the heavy science. Every article like this
needs some equations to make it look clever. So here are our equations.
First we make a force balance.

We found that the shape factor, k, was 1.130, just
a little greater than one, as expected. We found that the average drag coefficient
was 0.989 - a little over twice the drag coefficient of a heavy spheres
falling ( about 0.47).

We can solve these equations for the rise velocity
of the pibal in terms of the measured diameter and the weight of the empty
balloon. We solve these equations for "v" and get our magic formula.

We programmed the magic equation and solved it
for several diameters and several empty balloon weights. We graphed these
in Figure 1 and printed them in Table 1. Here is how you use Figure 1. Take
a bunch of the balloons you intend to use as pibals and weigh them. Say
eleven empty pibals weigh 30.8 grams. The average pibal weight is 30.8/11
= 2.8 grams (there are 28.4 grams in an ounce, if you have a scale that
weighs in ounces). Inflate your pibal with helium. Measure the diameter.
You can do this by using a carpenter's square as a caliper, measuring the
pibals shadow on a surface perpendicular to the sun, etc. Suppose your pibal
is 15 inches in diameter. To use the figure, go up the 15 inch line to about
half way between the 2 and 4 gram pibal lines and read about 415 ft/min.

To use the table, enter the column marked 3 grams.
Go down to the row marked 15 inches and read the value 416 ft/min. Our balloon
only weighed 2.8 grams so we go a little toward the 2 gram column (423 ft/min)
and estimate 418 ft/min. Don't waste your time trying to get closer than
about 5 ft/min.

If you want to use the magic formula, put everything
in consistent units. The factor of 60 in the magic formula converts from
ft/sec to ft/min. If the temperature or pressure is markedly different from
our standard conditions of 78· F and 750 mmHg you can change the
density with the following formula.